Polya's Theorem Research Articles

On a Two-Point Version of Transfinite Diameter

We study properties of a two-point version of the transfinite diameter of a set. By using relations obtained for its calculation, we prove a two-point version of the well-known Polya theorem on an estimate from above for the Hankel determinants of a holomorphic function.

A CENSUS OF TIGHT TRIANGULATIONS

A triangulation of a manifold (or pseudomanifold) is called a tight triangulation if any simplexwise linear embedding into any Euclidean space is tight. Tightness of an embedding means that the inclusion of any sublevel selected by a linear functional is injective in homology and, therefore, topologically essential. Tightness is a generalization of convexity, and the tightness of a triangulation is a fairly restrictive property. We give a review on all known examples of tight triangulations and formulate a (computer-aided) enumeration theorem for the case of at most 15 vertices and the presence of a vertex-transitive automorphism group. Altogether, six new examples of tight triangulations are presented, a vertex-transitive triangulation of the simply connected homogeneous 5-manifold SU(3)/SO(3) with vertex-transitive action, two non-symmetric 12-vertex triangulations of S3 × S2, and two non-symmetric triangulations of S3 × S3 on 13 vertices.

Asymptotic enumeration theorems for the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs

The asymptotic properties of the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs are studied. Let\(C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)\) be a directed circulant graph. Let\(\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)\) and\(\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)\) be the numbers of spanning trees and of Eulerian trails, respectively. Then $$\begin{array}{*{20}c} \begin{gathered} \lim \frac{1}{k}\sqrt[p]{{T\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)}} = 1, \hfill \\ \lim \frac{1}{{k!}}\sqrt[p]{{E\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)}} = 1, \hfill \\ \end{gathered} & {p \to \infty .} \\ \end{array} $$ Furthermore, their line digraph and iterations are dealt with and similar results are obtained for undirected circulant graphs.

Assessing robustness of crossover designsto subjects dropping out

In some crossover experiments, particularly in medical applications, subjects may fail to complete their sequences of treatments for reasons unconnected with the treatments received. A method is described of assessing the robustness of a planned crossover design, with more than two periods, to subjects leaving the study prematurely. The method involves computing measures of efficiency for every possible design that can result, and is therefore very computationally intensive. Summaries of these measures are used to choose between competing designs. The computational problem is reduced to a manageable size by a software implementation of Polya theory. The method is applied to comparing designs for crossover studies involving four treatments and four periods. Designs are identified that are more robust to subjects dropping out in the final period than those currently favoured in medical and clinical trials.

Application of the Sieve Method to the Enumeration of Stable Stereo and Position Isomers of a Series of Deoxycyclitols

A pattern inventory is developed for the enumeration of stable stereo and position isomers of any series of deoxycyclitols with the formula (CH2)p(CHOH)q where p(CH2) and q(CHOH) groups form a ring system of size n = (p + q). The counting method consists of two steps: firstly the application of POLYA's counting theorem gives the number of permutation isomers of the series; and secondly, the sequential enumeration of stable stereo and position isomers is carried out by applying Balasubramanian's procedure to each permutation isomer of the series. An illustration is shown in the series of deoxycycloheptitols where p + q = 7, 1 ≤ p ≤ 5 and 2 ≤ q ≤ 6.

Introduction to combinatorics

Introduction Permutations and Combinations The Inclusion-Exclusion Principle Partitions Stirling's Approximation Partitions and Generating Functions Generating Functions and Recurrence Relations Permutations and Groups Group Actions Graphs Counting Patterns Polya's Theorem Solutions to the Exercises Suggestions for Further Reading List of Symbols Index

Some enumeration theorems in pseudo-symplectic geometry over a finite field of characteristic two and the construction of association schemes

In this paper, we discuss the association schemes yielding subspaces of type ( m, 0, 0, 0) in the pseudo-symplectic geometry, and present some formulae of their class number and valency. Furthermore, the intersection numbers of the above association schemes have been computed when m = 2.

Forcing bonds of a benzenoid system

For a Kekulean benzenoid system, we can define the fixed single bonds, fixed double bonds and forcing bounds. The first and second types of bonds can be recognized by efficient algorithms. In this paper, we give an efficient algorithm to recognize the forcing bonds of a benzenoid system. For a cata-condensed benzenoid system we completely determine its forcing bonds. Furthermore, by Polya's theorem we enumerate all cata-condensed benzenoid Systems with forcing bonds.

Solar Cars and Variational Problems Equivalent to Shortest Paths

A classical theorem of Hardy, Littlewood, and Polya on rearrangements of functions is used to prove the equivalence of a class of variational problems. As a consequence, solutions are shortest paths and can be computed numerically via quadratic programming. Applications include solar cars and reservoirs.

A stabilization theorem for Hermitian forms and applications to holomorphic mappings

We consider positivity conditions both for real-valued functions of several complex variables and for Hermitian forms. We prove a stabilization theorem relating these two notions, and give some applications to proper mappings between balls in different dimensions. The technique of proof relies on the simple expression for the Bergman kernel function for the unit ball and elementary facts about Hilbert spaces. Our main result generalizes to Hermitian forms a theorem proved by Polya [HLP] for homogeneous real polynomials, which was obtained in conjunction with Hilbert's seventeenth problem. See [H] and [R] for generalizations of Polya's theorem of a completely different kind. The flavor of our applications is also completely different.

Comparison of determinantal inequalities for lower bounds to ?1/r?

This article considers two types of 2 × 2, 3 × 3, 4 × 4, and 5 × 5 determinantal inequalities. One type involves Gram determinants, while the other type involves determinants based on a theorem by Polya and Szego. The elements of the first type of determinants are 〈rn〉, while the elements of the second type of determinants are c 〈rn〉, where c = 3 + n, and r is the distance from the nucleus of an atom. Both types of determinantal inequalities are used to obtain lower-bound (LB) estimates of 〈1/r〉 for atoms of spherical symmetry. The inequalities involving Gram determinants have been applied previously by the author to atoms of spherically symmetric charge distributions, such as He, Ne, Ar, Kr, Xe; Li, Na, K, Rb; Be, Mg, Ca, Sr. In the resent article, the inequalities involving determinants based on the Polya and Szego theorem are applied to the same atoms. It is found that in both cases the LB values of 〈1/r〉, for all atoms considered, appear to converge to the quantum mechanical (QM) values of Boyd, who calculated them with the Roothan–Hartree–Fock wave functions of Clementi and Roetti. It is also found that the LB values of 〈1/r〉, using determinants based on the Polya–Szego theorem, converge “faster” to the QM values of 〈1/r〉 than do the LB values of 〈1/r〉 based on Gram determinants. © 1995 John Wiley & Sons, Inc.

The Lattice of Periods of a Group Action

Let L(G) denote the lattice of subgroups of G and Π(S) the lattice of partitions of S. Given an action of a finite group G on a finite set S, define the map η:L(G) → Π(S) by a, b ∈ S are in the same block of η(H) if and only if there is a g ∈ H such that g · a = b. The blocks of η(H) are the periods or orbits of the action of H on S. The image of η in Π(S) is a lattice called the lattice of periods of the group action and is denoted J(G, S). The lattice J(G, S) encodes a great deal of the information about the group action of G on S. Enough that a poset theoretic proof of Pólya′s Enumeration Theorem can be given using these lattices. [See G.-C. Rota, Bull. Amer. Math. Soc.75 (1969), 325-329 and 330-334; G.-C. Rota and D. Smith, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4)IV, No. 4 (1977).] This paper gives a systematic look at these lattices. The main result is that J(G,) depends only on the support of the representation of the action on the irreducible representations of G. If the representation of two group actions contain the same irreducible representations possibly with different multiplicities, then the resulting lattices of periods are the same. A method for generating all the possible lattices of periods for a particular group is also given.

Counting circular caterpillars

Our objects are (2.1) to introduce the graphical concept of a circular caterpillar together with cyclical code of non-negative integers, and (2.2) to provide an elementary example of the simplest kind of application of the celebrated Polya Enumeration Theorem.

An introduction to combinatorics, by Alan Slomson. Pp 270. £17.50. 1991. ISBN 0-412-35370-9 (Chapman and Hall)

Introduction Permutations and Combinations The Inclusion-Exclusion Principle Partitions Stirling's Approximation Partitions and Generating Functions Generating Functions and Recurrence Relations Permutations and Groups Group Actions Graphs Counting Patterns Polya's Theorem Solutions to the Exercises Suggestions for Further Reading List of Symbols Index

A note on the enumeration of equivalence classes in spectroscopy

In order to enumerate NMR signals and NQR spectral patterns, the number of equivalence classes under group action is to be determined by means of Polya's theorem in the literature. This note suggests a more straightforward approach by using Burnside's lemma.

Calculation of the indicator of an entire function of rational order in terms of its taylor coefficients

By using both the Polya theorem on the connection between the growth of an entire exponential function and the location of singularities of its Borel transform and the analog of this result for finite-order entire functions (due to Mclntyre), we obtain estimates for the indicator of the growth of an entire function in terms of its Taylor coefficients and, in some cases, determine this indicator exactly.

Two enumeration theorems in singular symplectic geometry and a class of PBIB designs

We give two enumeration theorems in singular symplectic geometry. Then, taking the subspaces of type (1,0,0) as the set of treatments, we construct a class of association schemes and PBIB designs with three associate classes and calculate their parameters.

Generalization of de Bruijn's extension of P�lya's theorem to all characters

We generalize the de Bruijn extension of Polya's theorem to all characters of finite permutation groups and show that the de Bruijn theorem becomes a special case of our generalization when applied to the character of the totally symmetric representation./ab]

A Note on Polya's Theorem

A Note on Polya's Theorem

The Richness of Structures Available to P2N2Inorganic Heterocycles: A Topological and Molecular' Orbital (EHMO) Analysis

Abstract A topological analysis, based upon Polya's theorem, was used to calculate all possible isomers of P2N2 ring derivatives, as a function of the number of substituents attached to phosphorus and nitrogen. The analysis considered four-, five- and six-coordinate phosphorus, and two-, three- and four-coordinate nitrogen. EHMO calculations established the HOMO-LUMO orbitals and the charges of variously substituted P2N2 heterocycles.